\\(> 1M<1:SKNTMENT AND PROOF IX (;KO^^■:TR^ 



Fig. 4 



These })roofs have been genetic rather than generic ; that is, 

 they show how the circle may be supposed to have been gene- 

 rated, but they do not place it in relation to the other circles 

 of the triangle. Yet even here we see a family of circles. The 

 orthocentre is ])erhaps the most fundamental point of the whole 

 triangle, and if the lines OA, OB. OC are divided in various 

 proportions and the points of division joined, we have an infinite 

 series of triangles with their circum-circles, such that O is the 

 centre of homology and of similitude and the orthocentre of 

 the whole group, and all the centres lie on ( )S. If J be the 

 moving ))oint. when the ratio ( )| : jS i^- n. we have Ihc starting 

 point; when it is i, the Nine-points Circle; when 2, the circle 

 whose centre is (i ; when oo^ the circum-circle. The family, 

 however, is not very important to the triangle, since the only 

 other obvious common propert)' seems to be that they cut ( )A, 

 OB, OC antiparallel to the oi)])osite sides. .^till, there is the 

 family, an infinite series of circles alternately Nine-])oinls and 

 Circum-circles to one another. 



Tile last proof was got by reversing the triangle on its 

 sides ; a far more fruitful ])rocess is to reverse it on its angles. 

 (See Figures 5 and 6.) The chief result of this is that all 

 transversals and all lines through the angles become antiparallel 

 to their former selves, and antiparallelism is a prolific source 

 of symmetry. After a diligent course of drilling with the aid 

 of tracing paper, even the most backward jnipils get a clear 

 notion of the properties of auti])arallels, c.(/. : — 



( I ) Every transversal with its antiparallel and the sides 

 makes a cyclic c|uadrilateral ; 



(2) Every ])arallel to a makes with its antiparallel the angle 

 B -- C, etc. ; 



(3) All antiparallels to the other two sides make the angle 

 A with a. B with b. C with c : therefore on each side of the 

 triangle there fall two sets of antiparallels, making isosceles 

 triangles on it. 



The drilling in this matter Ts so important that 1 add a 

 special exercise. Draw any angle BAC. and suppose a point 



